Non-orthogonal multiple access (NOMA) and device-to-device (D2D) are two key technologies of the fifth-generation wireless networks. In this paper, we propose a new D2D-and-NOMA integrated framework, where the D2D users (DUEs) can reuse the spectrum of the cellular users (CUEs) in four NOMA-aided spectrum-sharing modes. In order to fully exploit the potential of the proposed framework, we jointly optimize user pairing and power control to maximize the number of accessed D2D links and meanwhile reduce the total power consumption under the constraints of the decoding thresholds of the DUEs and CUEs. We first analytically obtain the optimal transmission power for each DUE-CUE pair. Then, based on the power control policy, we reformulate the user pairing problem as a min-cost max-flow problem in graph theory and solve it efficiently. Specifically, our proposed algorithm can solve the formulated problem optimally with low complexity. Finally, simulation results indicate that our algorithm can significantly improve the number of accessed D2D links and reduce the power consumption in comparison with the other schemes.